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Welcome to Algebra I: Introduction to Polynomials

22 minutes

Hey, guys. Welcome to Algebra I. Today's topic will introduce you to the world of polynomials. You'll apply some of what you learned in pre-algebra to help you with this one. Ready? Let's go.

(Describer) With a stylus, she activates a screen.

All right. In order to know exactly what a polynomial is, I have to take you back to what a monomial is. A polynomial is the sum of two or more monomials. But what's a monomial? Well, a monomial can be just a regular number, like 7 or 12 or a -4, or the fancy way to say a negative number in algebra is a constant term because it's value doesn't change. That 7 will constantly be 7. It will never change. That 12 is constantly 12, and that -4 is constantly -4. So, a monomial can be a constant term, or a regular number. Okay? A monomial can also be a variable, like x or v or y. Those aren't constant terms because their value can change, so we call them variables. A monomial could be a variable. A monomial could also be the product of a constant and a variable. For example, 5x or 7b-squared, -12cd. Those are also monomials. Basically, a monomial, in general, is one term. Either a constant term, a variable, or the product of a constant and a variable. All of those are considered monomials. A binomial is a polynomial that consists of two terms. For example, 4x plus 1. I've added two terms together. It's called a binomial, is its more specific name. 3b-squared minus 1? That's a binomial, X plus 3y. That's a binomial. A binomial is a polynomial that only consists of two terms. There's a lot of tongue-twisters on this one. Okay. A trinomial-- I bet you might have noticed from that prefix "tri"-- is a polynomial with three terms. For example, 5x-squared plus 2x plus 4. Or, here we go, 4a-cubed b-squared plus 6b minus 1. That's a trinomial. It's the sum of three terms. Or x-cubed minus 8x plus 12. That's a trinomial. A trinomial is a polynomial with three terms. You can think of polynomial as the name that encompasses the big group. But more specifically, you could have a monomial, which is just one term; a binomial, which is the sum of two terms; or a trinomial, which is the sum of three terms. Collectively, we refer to them as polynomials. "Poly," that prefix-- it just means many. Polynomial is the sum of many terms.

(Describer) She changes the screen to show Example 1.

Okay. One kind of problem you'll run into in the world that is polynomials is you have to find the degree of the polynomial. How you find the degree of a polynomial is you have to zone in on the exponents. Your largest exponent will be the degree of the polynomial. The value of that largest exponent is the value of the degree of your polynomial. We've got to go term by term to see what exactly is the degree of each term. Bear with me for a second. Get that pen.

(female describer) 5x-squared plus 3x-squared y to the 4th power minus 6.

(Describer) 5x-squared plus 3x-squared y to the fourth power minus 6.

So, for 5x-squared,

(Describer) She writes.

my exponent in that first term is a 2. So, I can say the degree of that first term is 2. All righty? Now, for the second term-- Yeah, let's go underneath. 3x-squared y to the 4th. In that term, I have two exponents. When you have more than one exponent in a term, just add your exponents together. Here I have a 2 and a 4. So, I'd say the degree of this term is 2 plus 4, which is 6. Right? So far, it's looking like the largest degree I've got going is 6. Look at that -6, that constant term. We've got to throw back to your knowledge of exponents for a bit. You remember that when you're dealing with exponents, for example, I can raise anything to the 0 power and it'll equal 1? Right? So, I could think about this as -6 times x to the zero power, because essentially it's the same thing as -6 times 1. I'm not changing the value of that constant term. But knowing that and looking at it this way will help you figure out the degree of that constant term. Now you've introduced an exponent to that term without changing its value. So, now I can see, well, okay, the exponent 0. So, the degree of that constant term, that -6, is 0. Or a shortcut is that you could think about it like every constant term will always have a degree of 0. You can treat every single constant term in that same fashion. For terms where you actually see exponents, just take them. For 5x-squared, the degree was 2. For that 3x-squared y to the 4th, I had to add those exponents together. So, the degree was 6. And then for my constant term, that -6, the degree is 0. Looking at all of your degrees here, you have a 2 right here, you've got the 6, and you have the 0. Out of 2 and 6 and 0, the biggest value is 6. So, I'd say the degree of my polynomial is 6. Because it's whatever that largest degree is amongst all your terms. Let's make sure you've got the hang of that. So, "What is the degree of the polynomial?" 8x-squared z-cubed plus 5z-squared minus 1. Let's find the degree of each of these terms. Gotta shift that problem up, give myself more room. All righty. For my first term, I have 8x-squared z-cubed. I have more than one exponent here. I need to add the exponents together to get the degree of that term. So, 2 plus 3, that's 5. So, I'd say that first term has a degree of 5. All righty? Let's look at that second term: 5z-squared. I've only got one exponent there, so that's simple. The degree is 2. My only exponent there is 2. Then I see I have this -1 here. That's a constant term. You remember the shortcut that we established on that last one? Every single constant term has a degree of 0, and we saw that by looking at how we could represent that product, a constant term times x to the zero power. That constant term here, -1, has a degree of 0. Check out all your degrees. Which one's the biggest? The 5. I'd say this polynomial has a degree of 5.

(Describer) She writes "degree 5".

All right. Let's keep going. "What is the degree of the polynomial?" -7a-cubed b plus 2a-cubed plus b-squared. So, take each term separately, each monomial separately. All righty, so you have -7a-cubed b. You remember, with exponents, when you don't see one-- like this term here, or this part here, this b-- there's a 1 I'm not seeing. I have a sum of exponents here, 3 plus 1. I'll have to deal with that. So, 3 plus 1, that's 4, So, this monomial has a degree of 4. All righty. Now I've got my 2a-cubed. The 2a-cubed, I only see this one exponent here. That's simple. This one has a degree of 3. All righty. Then that last monomial, b-squared, there's just one exponent in this one. So, the degree is 2. Then check out all your degrees. You've got 4, you've got 3, you've got 2. The biggest one is 4. So, the degree of this polynomial...is 4. All right. Let's keep going. And I do believe it is your turn. Press pause for a bit, work through this problem, and when you're ready to compare answers with me, press play.

(Describer) Title: What is the degree of the polynomial? 9x to the fifth power y minus 4x cubed plus 2-squared.

(describer) "What is the degree of the polynomial?" 9x to the 5th power y minus 4x-cubed plus x-squared.

(instructor) Ready to check? Let's see. Let's get the pointer. Move it out of the way. All righty. 9x to the 5th y minus 4x-cubed plus x-squared. What's the degree? It is 6. Okay? If you need to see how I got that, this is how I got it. Let's get the pen. So, I took each term individually. So, 9x to the 5th y. I didn't see an exponent with that y. That told me there was a 1 I wasn't seeing. I have more than one exponent, so I need to add them together. So, 5 plus 1, that's 6. I knew that first term had a degree of 6. Then I took my next term, -4x-cubed. I only had one exponent to focus on there, the 3. So, this term has a degree of 3. Then I have the last term, that x-squared. Only one exponent to focus on, and it's a 2. So, your degree of that term is 2. Then check out all your degrees. Which one's the biggest? In this case, the 6. So, the degree of the polynomial was 6. That's how I got that one. Okay? Let's keep going. A little more to introduce you to polynomials. You may also run into problems that ask you to find the leading coefficient.

(Describer) Example 4: 8 x-squared minus two x plus seven x-cubed minus 4.

(describer) Example 4: 8x-squared minus 2x plus 7x-cubed minus 4.

(instructor) How you do that is you first have to make sure that your polynomial is written in standard form. Standard form means that your polynomial goes from the term with the largest exponent, down to the term with the smallest exponent. That's how you write it in standard form. Sometimes the polynomial will be given to you in standard form. Then you just pick out your leading coefficient. But sometimes it's not given to you in standard form. You've got to do maneuvering to get it there. Looking at this polynomial, I see it's not written in standard form. My largest exponent is a 3, and that 7x-squared is not written first. I need to rearrange this. All righty. So, I'm gonna write this as 7x-cubed, 'cause I'm gonna take this first term first. You notice how I'm circling the plus sign in front of it? That lets me know what the sign of my term is. I know this 7x-cubed is a positive 7x-cubed, and it has to be out front if I write it in standard form, 'cause it has the largest exponent. So, 7x-cubed. I've handled it, so I'm gonna cross it out. I've got an exponent of 3. I see I have an exponent of 2, this 8x-squared. There's no sign out front. You know from pre-algebra that means it's positive. Okay. So, I can write this as 7x-cubed plus 8x-squared. All righty? And I've handled that. I've got an exponent of 3, exponent of 2, and you have an exponent of 1. That negative 2x, there's a 1 there that you're not seeing. Gonna circle that term. All righty. You notice I circled that subtraction out front? That subtraction tells me that 2x is negative. When I'm rewriting my polynomial, I'm gonna write -2x. Minus 2x, negative 2x. They're kind of interchangeable. You can think about them the same way. I've handled that. I see I have this constant term at the end, this -4. So, at the end here: -4. Now this polynomial is written in standard form. The exponents go from 3, 2, 1, and then a 0. As you remember, I can think about that -4 as a -4x to the 0.

(describer) She's written 7x-cubed

(Describer) She's written 7x-cubed plus 8 x-squared minus 2x minus 4.

plus 8x-squared minus 2x minus 4. They don't always go sequential like this, an exponent of 3, an exponent of 2, an exponent of 1. Sometimes they skip one, but you want to make sure that those exponents go in descending order. Now that this is written in standard form, I can figure out what the leading coefficient is. The leading coefficient is the coefficient of your first term. The leading term is coefficient. My first term is 7x-cubed, and the coefficient there is 7. So, the leading coefficient of this polynomial is 7. And you're all done. It took a bit of work to get there, but the answer is that constant, that 7. Let's keep going. What's the leading coefficient of this polynomial? I see I have 3x minus x to the 4th plus 5x-squared. I have to do some maneuvering, because it's not in standard form. I know that because I see that that middle term has the largest exponent, 4, and it's not written first. And when polynomials are written in standard form, it's gonna start from the largest and go down the line. I need to do some rearranging. That -x to the 4th needs to be written first. So, -x to the 4th. All right, handled that. I don't have an exponent of 3, but I do have an exponent of 2. It's attached to this term here. That's a positive 5x-squared. So, plus 5x-squared. Handled that. Then my last term is this 3x. There's no sign out front of it. That tells me it's positive. I can write that as plus 3x. Now that this polynomial is written in standard form, now you can pick out the leading coefficient. Your first term is -x to the 4th, and same for coefficients as exponents. When you don't see a coefficient, there is an invisible one there. So, this first term is like -1x to the 4th, which would mean that its coefficient is -1. My leading coefficient for this polynomial is -1. Okay? Let's try one more: 2 plus 9x. All right. That's not in standard form. How do I know that? Because that x has an exponent of 1. This 2 is a constant term. I know there's an x to the 0 there. So, this isn't written in the right order. This term should be first. It has the largest exponent. So, 9x--It's positive, but you don't have to write the plus when it's the first term. I've handled that. Then you have the 2. There's no sign out front, so it's positive. This really should be written as 9x plus 2 for it to be in standard form. Now that it's in standard form, look at your leading term. It's 9x, and the coefficient of that first term is 9. So, the leading coefficient of that polynomial is 9. All right? All right, it's your turn. Go ahead and press pause, take a minute, and work through these problems. Remember how we did it. First, make sure that polynomial is written in standard form. Then pick out the leading coefficient. When you're ready to compare answers with me, press play.

(Describer) Title: What is the leading coefficient? Number One: 3 plus 4x minus 9x-cubed plus 17x-squared Number Two: 11 minus 3x-squared plus x to the fourth.

(describer) What is the leading coefficient? Number 1: 3 plus 4x minus 9x-cubed plus 17x-squared. Number 2: 11 minus 3x-squared plus x to the fourth. You ready to check? Let's check. Let's get the pointer tool. For that first polynomial, 3 plus 4x minus 9x-cubed plus 17x-squared, the leading coefficient: -9. For the second one, 11 minus 3x-squared plus x to the 4th, that leading coefficient is 1. If you need to see how I did it, here we go.

(Describer) She switches to just the first problem.

So, first, I noticed this wasn't written in standard form, because the term with the largest exponent wasn't written first. I spotted that three, and I knew it was a part of that term. So, that should have been written first: -9x-cubed. All right. Then I handled that. I saw I had an exponent of 2, and that's the next one, going in descending order. That's a positive 17x-squared. So, plus 17x-squared. Now I handled that. Now I see the 4x. It's positive. So, plus 4x. All righty. Then I see the 3. There's no sign in front of it, so it's positive. So, plus 3. Now that it's written in standard form, look at your first coefficient-- or look at your first term. It's -9x-cubed, and the coefficient of that term is -9. So, the leading coefficient is -9. Okay? Need to see the second one?

(Describer) She switches to it.

All righty. 11 minus 3x-squared plus x to the 4th. This wasn't written in standard form, so I had to rearrange it first. That 4 is the largest exponent. That means that x to the 4th should be written first. It's positive. So, x to the 4th. Right? Handled that. Next exponent, going in descending order, would be 2, and it's attached to that term. So, -3x-squared. Okay. Now I've handled that. And that constant term, 11, there's no sign in front of it, so it's positive. So, plus 11. Right? Then I gave it a glance over. I see that my first term is x to the 4th. I don't see a coefficient, so I know there's an invisible 1. So, my leading coefficient is 1. All right. Hope you're getting comfortable with the world of polynomials. We'll do more with this topic, but I hope that you got a good introduction. See you soon. Bye.

(Describer) Accessibility provided by the US Department of Education.

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In this program, students explore the properties of polynomials. A polynomial is the sum of one or more terms. In their simplest form, they contain no like terms, and they are written in descending order. Part of the "Welcome to Algebra I" series.

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Runtime: 22 minutes

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